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''Kummer function, Pochhammer function''
 
''Kummer function, Pochhammer function''
  
 
A solution of the [[Confluent hypergeometric equation|confluent hypergeometric equation]]
 
A solution of the [[Confluent hypergeometric equation|confluent hypergeometric equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
zw  ^ {\prime\prime} + ( \gamma - z) w  ^  \prime  - \alpha w  = 0.
 +
$$
  
 
The function may be defined using the so-called Kummer series
 
The function may be defined using the so-called Kummer series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\Phi ( \alpha ;  \gamma ;  z)  = \
 +
{} _ {1} F _ {1} ( \alpha , \gamma ; z) =
 +
$$
 +
 
 +
$$
 +
= \
 +
1 + {
 +
\frac \alpha  \gamma
 +
} {
 +
\frac{z}{1!}
 +
} +
 +
\frac{
 +
\alpha ( \alpha + 1) }{\gamma ( \gamma + 1) }
 +
 +
\frac{z  ^ {2} }{2! }
 +
+ \dots ,
 +
$$
 +
 
 +
where  $  \alpha $
 +
and  $  \gamma $
 +
are parameters which assume any real or complex values except for  $  \gamma = 0, - 1, - 2 \dots $
 +
and  $  z $
 +
is a complex variable. The function  $  \Psi ( \alpha ; \gamma ; z ) $
 +
is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247003.png" /></td> </tr></table>
+
$$
 +
\Psi ( \alpha ; \gamma ; z)  = \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247005.png" /> are parameters which assume any real or complex values except for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247007.png" /> is a complex variable. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247008.png" /> is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),
+
\frac{\Gamma ( \alpha - \gamma + 1) \Gamma ( \gamma - 1) }{\Gamma ( \alpha ) \Gamma ( 1 - \gamma ) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c0247009.png" /></td> </tr></table>
+
z ^ {1 - \gamma } \Phi ( \alpha - \gamma + 1 ; 2 - \gamma ; z),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470010.png" /></td> </tr></table>
+
$$
 +
\gamma  \neq  0 , - 1 , - 2 \dots \  |  \mathop{\rm arg}  z |  < \pi ,
 +
$$
  
 
is called the confluent hypergeometric function of the second kind.
 
is called the confluent hypergeometric function of the second kind.
  
The confluent hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470011.png" /> is an entire analytic function in the entire complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470012.png" />-plane; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470013.png" /> is fixed, it is an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470014.png" /> and a meromorphic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470015.png" /> with simple poles at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470016.png" />. The confluent hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470017.png" /> is an analytic function in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470018.png" />-plane with the slit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470019.png" /> and an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470021.png" />.
+
The confluent hypergeometric function $  \Phi ( \alpha ;  \gamma ;  z ) $
 +
is an entire analytic function in the entire complex $  z $-
 +
plane; if $  z $
 +
is fixed, it is an entire function of $  \alpha $
 +
and a meromorphic function of $  \gamma $
 +
with simple poles at the points $  \gamma = 0, - 1 , - 2 ,\dots $.  
 +
The confluent hypergeometric function $  \Psi ( \alpha ;  \gamma ;  z ) $
 +
is an analytic function in the complex $  z $-
 +
plane with the slit $  ( - \infty , 0 ) $
 +
and an entire function of $  \alpha $
 +
and $  \gamma $.
  
The confluent hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470022.png" /> is connected with the [[Hypergeometric function|hypergeometric function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470023.png" /> by the relation
+
The confluent hypergeometric function $  \Phi ( \alpha ;  \gamma ;  z ) $
 +
is connected with the [[Hypergeometric function|hypergeometric function]] $  F ( \alpha , \beta , \gamma ;  z ) $
 +
by the relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470024.png" /></td> </tr></table>
+
$$
 +
\Phi ( \alpha ;  \gamma ; z)  = \
 +
\lim\limits _ {\beta \rightarrow \infty }  F
 +
\left (  \alpha , \beta , \gamma ; \
 +
{
 +
\frac{z} \beta
 +
} \right ) .
 +
$$
  
Elementary relationships. The four functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470026.png" /> are called adjacent (or contiguous) to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470027.png" />. There is a linear relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470028.png" /> and any two functions adjacent to it, e.g.
+
Elementary relationships. The four functions $  \Phi ( \alpha \pm  1 ;  \gamma ;  z ) $,  
 +
$  \Phi ( \alpha ;  \gamma \pm  1 ;  z ) $
 +
are called adjacent (or contiguous) to the function $  \Phi ( \alpha ;  \gamma ;  z ) $.  
 +
There is a linear relationship between $  \Phi ( \alpha ;  \gamma ;  z ) $
 +
and any two functions adjacent to it, e.g.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470029.png" /></td> </tr></table>
+
$$
 +
\gamma \Phi ( \alpha ;  \gamma ;  z) -
 +
\gamma \Phi ( \alpha - 1 ; \gamma ; z) -
 +
z \Phi ( \alpha ; \gamma + 1 ;  z)  = 0.
 +
$$
  
Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470030.png" /> with the associated functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470033.png" /> are integers.
+
Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function $  \Phi ( \alpha ;  \gamma ;  z ) $
 +
with the associated functions $  \Phi ( \alpha + m ;  \gamma + n ;  z) $,  
 +
where $  m $
 +
and $  n $
 +
are integers.
  
 
Differentiation formulas:
 
Differentiation formulas:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470034.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470035.png" /></td> </tr></table>
+
\frac{d  ^ {n} }{dz  ^ {n} }
 +
 
 +
\Phi ( \alpha ;  \gamma ; z)  = \
 +
 
 +
\frac{\alpha \dots ( \alpha + n - 1 ) }{\gamma \dots ( \gamma + n - 1 ) }
 +
 
 +
\Phi ( \alpha + n ; \gamma + n ; z),
 +
$$
 +
 
 +
$$
 +
= 1 , 2 , .  . . .
 +
$$
  
 
Basic integral representations.
 
Basic integral representations.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470036.png" /></td> </tr></table>
+
$$
 +
\Phi ( \alpha ; \gamma ; z) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470037.png" /></td> </tr></table>
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470038.png" /></td> </tr></table>
+
\frac{\Gamma ( \gamma ) }{\Gamma ( \alpha ) \Gamma ( \gamma - \alpha
 +
) }
 +
\int\limits _ { 0 } ^ { 1 }  e ^ {zt } t ^ {\alpha - 1
 +
} ( 1 - t) ^ {\gamma - \alpha - 1 }  dt ,\  \mathop{\rm Re}  \gamma > \mathop{\rm Re}  \alpha > 0 ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470039.png" /></td> </tr></table>
+
$$
 +
\Psi ( \alpha ; \gamma ; z) =
 +
$$
  
The asymptotic behaviour of confluent hypergeometric functions as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470040.png" /> can be studied using the integral representations [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470041.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470043.png" /> are bounded, the behaviour of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470044.png" /> is described by formula (2). In particular, for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470045.png" /> and bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470047.png" />:
+
$$
 +
= \
 +
{
 +
\frac{1}{\Gamma ( \alpha ) }
 +
} \int\limits _ { 0 } ^  \infty  e ^ {- zt } t ^
 +
{\alpha - 1 } ( 1 + t) ^ {\gamma - \alpha - 1 }  dt ,\  \mathop{\rm Re}  \alpha > 0 ,\  \mathop{\rm Re}  z > 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470048.png" /></td> </tr></table>
+
The asymptotic behaviour of confluent hypergeometric functions as  $  z \rightarrow \infty $
 +
can be studied using the integral representations [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]. If  $  \gamma \rightarrow \infty $,
 +
while  $  \alpha $
 +
and  $  z $
 +
are bounded, the behaviour of the function  $  \Phi ( \alpha ;  \gamma ; z) $
 +
is described by formula (2). In particular, for large  $  \gamma $
 +
and bounded  $  \alpha $
 +
and  $  z $:
 +
 
 +
$$
 +
\Phi ( \alpha ; \gamma ; z)  = \
 +
1 + O ( | \gamma |  ^ {-} 1 ) .
 +
$$
  
 
Representations of functions by confluent hypergeometric functions.
 
Representations of functions by confluent hypergeometric functions.
Line 55: Line 166:
 
Bessel functions:
 
Bessel functions:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470049.png" /></td> </tr></table>
+
$$
 +
J _  \nu  ( z)  = \
 +
{
 +
\frac{1}{\Gamma ( 1 + \nu ) }
 +
}
 +
\left ( {
 +
\frac{z}{2}
 +
} \right )  ^  \nu
 +
e ^ {- iz } \Phi \left (
 +
\nu + {
 +
\frac{1}{2}
 +
} ; 2 \nu + 1 ; 2iz \right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470050.png" /></td> </tr></table>
+
$$
 +
I _  \nu  ( z)  = {
 +
\frac{1}{\Gamma ( 1 + \nu ) }
 +
} \left (
 +
{
 +
\frac{z}{2}
 +
} \right )  ^  \nu  e ^ {- z } \Phi \left ( \nu +
 +
{
 +
\frac{1}{2}
 +
} ; 2 \nu + 1 ; 2z \right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470051.png" /></td> </tr></table>
+
$$
 +
K _  \nu  ( z)  = \sqrt \pi e ^ {- z } ( 2z)  ^  \nu  \Psi \left (
 +
\nu + {
 +
\frac{1}{2}
 +
} ; 2 \nu + 1 ; 2z \right ) .
 +
$$
  
 
Laguerre polynomials:
 
Laguerre polynomials:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470052.png" /></td> </tr></table>
+
$$
 +
L _ {n} ^ {( \alpha ) } ( z)  = \
 +
 
 +
\frac{( \alpha + 1) _ {n} }{n! }
 +
\Phi  (- n ; \alpha + 1 ; z).
 +
$$
  
 
Probability integrals:
 
Probability integrals:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470053.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm erf} ( z)  = \
 +
 
 +
\frac{2z }{\sqrt \pi }
 +
\Phi
 +
\left ( {
 +
\frac{1}{2}
 +
} ; {
 +
\frac{3}{2}
 +
} ; - z  ^ {2} \right ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470054.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm erf}  c ( z)  = {
 +
\frac{1}{\sqrt \pi}
 +
} e ^ {- x  ^ {2} } \Psi
 +
\left ( {
 +
\frac{1}{2}
 +
} ; {
 +
\frac{1}{2}
 +
} ; z  ^ {2} \right ) .
 +
$$
  
 
The exponential integral function:
 
The exponential integral function:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470055.png" /></td> </tr></table>
+
$$
 +
-  \mathop{\rm Ei} (- z)  = \
 +
e  ^ {-} z \Psi ( 1 ; 1 ; z) .
 +
$$
  
 
The [[logarithmic integral]] function:
 
The [[logarithmic integral]] function:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470056.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm li} ( z)  = \
 +
z \Psi ( 1 ; 1 ; - \mathop{\rm ln}  z) .
 +
$$
  
 
Gamma-functions:
 
Gamma-functions:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470057.png" /></td> </tr></table>
+
$$
 +
\Gamma ( \alpha , z)  = \
 +
e ^ {- z } \Psi ( 1 - \alpha ; 1 - \alpha ; z) .
 +
$$
  
 
Elementary functions:
 
Elementary functions:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470058.png" /></td> </tr></table>
+
$$
 +
e  ^ {z}  = \Phi ( \alpha ; \alpha ; z) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024700/c02470059.png" /></td> </tr></table>
+
$$
 +
\sin  z  = e ^ {iz } z \Phi ( 1 ; 2 ; - 2iz) .
 +
$$
  
 
See also [[#References|[1]]], [[#References|[2]]], [[#References|[3]]], [[#References|[8]]].
 
See also [[#References|[1]]], [[#References|[2]]], [[#References|[3]]], [[#References|[8]]].

Latest revision as of 17:46, 4 June 2020


Kummer function, Pochhammer function

A solution of the confluent hypergeometric equation

$$ \tag{1 } zw ^ {\prime\prime} + ( \gamma - z) w ^ \prime - \alpha w = 0. $$

The function may be defined using the so-called Kummer series

$$ \tag{2 } \Phi ( \alpha ; \gamma ; z) = \ {} _ {1} F _ {1} ( \alpha , \gamma ; z) = $$

$$ = \ 1 + { \frac \alpha \gamma } { \frac{z}{1!} } + \frac{ \alpha ( \alpha + 1) }{\gamma ( \gamma + 1) } \frac{z ^ {2} }{2! } + \dots , $$

where $ \alpha $ and $ \gamma $ are parameters which assume any real or complex values except for $ \gamma = 0, - 1, - 2 \dots $ and $ z $ is a complex variable. The function $ \Psi ( \alpha ; \gamma ; z ) $ is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),

$$ \Psi ( \alpha ; \gamma ; z) = \ \frac{\Gamma ( \alpha - \gamma + 1) \Gamma ( \gamma - 1) }{\Gamma ( \alpha ) \Gamma ( 1 - \gamma ) } z ^ {1 - \gamma } \Phi ( \alpha - \gamma + 1 ; 2 - \gamma ; z), $$

$$ \gamma \neq 0 , - 1 , - 2 \dots \ | \mathop{\rm arg} z | < \pi , $$

is called the confluent hypergeometric function of the second kind.

The confluent hypergeometric function $ \Phi ( \alpha ; \gamma ; z ) $ is an entire analytic function in the entire complex $ z $- plane; if $ z $ is fixed, it is an entire function of $ \alpha $ and a meromorphic function of $ \gamma $ with simple poles at the points $ \gamma = 0, - 1 , - 2 ,\dots $. The confluent hypergeometric function $ \Psi ( \alpha ; \gamma ; z ) $ is an analytic function in the complex $ z $- plane with the slit $ ( - \infty , 0 ) $ and an entire function of $ \alpha $ and $ \gamma $.

The confluent hypergeometric function $ \Phi ( \alpha ; \gamma ; z ) $ is connected with the hypergeometric function $ F ( \alpha , \beta , \gamma ; z ) $ by the relation

$$ \Phi ( \alpha ; \gamma ; z) = \ \lim\limits _ {\beta \rightarrow \infty } F \left ( \alpha , \beta , \gamma ; \ { \frac{z} \beta } \right ) . $$

Elementary relationships. The four functions $ \Phi ( \alpha \pm 1 ; \gamma ; z ) $, $ \Phi ( \alpha ; \gamma \pm 1 ; z ) $ are called adjacent (or contiguous) to the function $ \Phi ( \alpha ; \gamma ; z ) $. There is a linear relationship between $ \Phi ( \alpha ; \gamma ; z ) $ and any two functions adjacent to it, e.g.

$$ \gamma \Phi ( \alpha ; \gamma ; z) - \gamma \Phi ( \alpha - 1 ; \gamma ; z) - z \Phi ( \alpha ; \gamma + 1 ; z) = 0. $$

Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function $ \Phi ( \alpha ; \gamma ; z ) $ with the associated functions $ \Phi ( \alpha + m ; \gamma + n ; z) $, where $ m $ and $ n $ are integers.

Differentiation formulas:

$$ \frac{d ^ {n} }{dz ^ {n} } \Phi ( \alpha ; \gamma ; z) = \ \frac{\alpha \dots ( \alpha + n - 1 ) }{\gamma \dots ( \gamma + n - 1 ) } \Phi ( \alpha + n ; \gamma + n ; z), $$

$$ n = 1 , 2 , . . . . $$

Basic integral representations.

$$ \Phi ( \alpha ; \gamma ; z) = $$

$$ = \ \frac{\Gamma ( \gamma ) }{\Gamma ( \alpha ) \Gamma ( \gamma - \alpha ) } \int\limits _ { 0 } ^ { 1 } e ^ {zt } t ^ {\alpha - 1 } ( 1 - t) ^ {\gamma - \alpha - 1 } dt ,\ \mathop{\rm Re} \gamma > \mathop{\rm Re} \alpha > 0 ; $$

$$ \Psi ( \alpha ; \gamma ; z) = $$

$$ = \ { \frac{1}{\Gamma ( \alpha ) } } \int\limits _ { 0 } ^ \infty e ^ {- zt } t ^ {\alpha - 1 } ( 1 + t) ^ {\gamma - \alpha - 1 } dt ,\ \mathop{\rm Re} \alpha > 0 ,\ \mathop{\rm Re} z > 0 . $$

The asymptotic behaviour of confluent hypergeometric functions as $ z \rightarrow \infty $ can be studied using the integral representations [1], [2], [3]. If $ \gamma \rightarrow \infty $, while $ \alpha $ and $ z $ are bounded, the behaviour of the function $ \Phi ( \alpha ; \gamma ; z) $ is described by formula (2). In particular, for large $ \gamma $ and bounded $ \alpha $ and $ z $:

$$ \Phi ( \alpha ; \gamma ; z) = \ 1 + O ( | \gamma | ^ {-} 1 ) . $$

Representations of functions by confluent hypergeometric functions.

Bessel functions:

$$ J _ \nu ( z) = \ { \frac{1}{\Gamma ( 1 + \nu ) } } \left ( { \frac{z}{2} } \right ) ^ \nu e ^ {- iz } \Phi \left ( \nu + { \frac{1}{2} } ; 2 \nu + 1 ; 2iz \right ) , $$

$$ I _ \nu ( z) = { \frac{1}{\Gamma ( 1 + \nu ) } } \left ( { \frac{z}{2} } \right ) ^ \nu e ^ {- z } \Phi \left ( \nu + { \frac{1}{2} } ; 2 \nu + 1 ; 2z \right ) , $$

$$ K _ \nu ( z) = \sqrt \pi e ^ {- z } ( 2z) ^ \nu \Psi \left ( \nu + { \frac{1}{2} } ; 2 \nu + 1 ; 2z \right ) . $$

Laguerre polynomials:

$$ L _ {n} ^ {( \alpha ) } ( z) = \ \frac{( \alpha + 1) _ {n} }{n! } \Phi (- n ; \alpha + 1 ; z). $$

Probability integrals:

$$ \mathop{\rm erf} ( z) = \ \frac{2z }{\sqrt \pi } \Phi \left ( { \frac{1}{2} } ; { \frac{3}{2} } ; - z ^ {2} \right ) , $$

$$ \mathop{\rm erf} c ( z) = { \frac{1}{\sqrt \pi} } e ^ {- x ^ {2} } \Psi \left ( { \frac{1}{2} } ; { \frac{1}{2} } ; z ^ {2} \right ) . $$

The exponential integral function:

$$ - \mathop{\rm Ei} (- z) = \ e ^ {-} z \Psi ( 1 ; 1 ; z) . $$

The logarithmic integral function:

$$ \mathop{\rm li} ( z) = \ z \Psi ( 1 ; 1 ; - \mathop{\rm ln} z) . $$

Gamma-functions:

$$ \Gamma ( \alpha , z) = \ e ^ {- z } \Psi ( 1 - \alpha ; 1 - \alpha ; z) . $$

Elementary functions:

$$ e ^ {z} = \Phi ( \alpha ; \alpha ; z) , $$

$$ \sin z = e ^ {iz } z \Phi ( 1 ; 2 ; - 2iz) . $$

See also [1], [2], [3], [8].

References

[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1980) (Translated from Russian)
[3] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1964)
[4] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[5] A.L. Lebedev, R.M. Fedorova, "Handbook of mathematical tables" , Moscow (1956) (In Russian)
[6] N.M. Burunova, "Handbook of mathematical tables" , Moscow (1959) (In Russian)
[7] A.A. Fletcher, J.C.P. Miller, L. Rosenhead, L.J. Comrie, "An index of mathematical tables" , 1–2 , Oxford Univ. Press (1962)
[8] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
How to Cite This Entry:
Confluent hypergeometric function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Confluent_hypergeometric_function&oldid=46450
This article was adapted from an original article by E.A. Chistova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article